Compact coalgebras, compact quantum groups and the positive antipode

Details Compact coalgebras, compact quantum groups and the positive antipode Compact coalgebras, compact quantum groups and the positive antipode

2009-07-07

arxiv.org/abs/0907.1183v1

Sao Paulo Journal of Mathematical Sciences, 3 (1), 2009, pp 191-227

Mathematics

Quantum Algebra

Scientific paper

In this article -that has also the intention to survey some known results in the theory of compact quantum groups using methods different from the standard and with a strong algebraic flavor- we consider compact o-coalgebras and Hopf algebras. In the case of a o-Hopf algebra we present a proof of the characterization of the compactness in terms of the existence of a positive definite integral, and use our methods to give an elementary proof of the uniqueness - up to conjugation by an automorphism of Hopf algebras- of the compact involution appearing in [4]. We study the basic properties of the positive square root of the antipode square that is a Hopf algebra automorphism that we call the positive antipode. We use it -as well as the unitary antipode and the Nakayama automorphism- in order to enhance our understanding of the antipode itself.

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